WASHINGTON—The House narrowly passed a measure approving a $3.5 trillion budget blueprint and locking in a late September vote on a roughly $1 trillion infrastructure bill, ending a standoff between centrist Democrats and party leaders over their legislative agenda.

Tuesday’s passage of the $3.5 trillion budget framework in a 220-212 vote unlocks a process, known as reconciliation, allowing Democrats to pass a broad package of healthcare, education and climate provisions in the Senate without GOP support, so long as all 50 senators in the Democratic caucus back it. The Senate passed the same budget blueprint earlier this month, and lawmakers are already working on drafting detailed legislation.

The measure passed after concessions from both House Speaker Nancy Pelosi (D., Calif.) and a group of 10 centrist Democrats. The centrists had balked at approving the budget framework and urged the House to vote immediately on the infrastructure bill, which has already passed the Senate.

Although the centrists didn’t secure an immediate vote on the infrastructure measure, they did pin down a Sept. 27 deadline for it, which means the House could vote on that bill weeks before the budget package is ready. Mrs. Pelosi has said previously that the House wouldn’t vote on infrastructure until the healthcare and education package has passed the Senate.

“This agreement does what we set out to do: secure a stand-alone vote for the bipartisan infrastructure bill, send it to the president’s desk and then separately consider the reconciliation package,” the group of centrists said Tuesday.